Question

the following rational equation has denominators that contain variables. for this equation, a. write the value or values of the variable that make a denominator zero. these are the restrictions on the variable. b. keeping the restrictions in mind, solve the equation.\\(\frac{2x}{x + 3}=5-\frac{6}{x + 3}\\)\
a. what are the value or values of the variable that makes the denominators zero?\\(x = -3\\)\
(simplify your answer. use a comma to separate answers as needed.)\
b. solve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your choice\
\\(\bigcirc\\) a. the solution set is \\(\\{\\ \\}\\).\
(type an integer or a simplified fraction. use a comma to separate answers as needed.)\
\\(\bigcirc\\) b. the solution set is \\(\\{x|x\\) is a real number\\(\\}\\).\
\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Find restriction on x

Set denominator to zero: $x+3=0$
Solve for x: $x=-3$

Step2: Eliminate denominators

Multiply all terms by $x+3$ (where $x
eq-3$):
$$2x = 5(x+3) - 6$$

Step3: Expand right-hand side

$$2x = 5x + 15 - 6$$

Step4: Simplify right-hand side

$$2x = 5x + 9$$

Step5: Isolate x terms

Subtract $5x$ from both sides:
$$2x - 5x = 9$$
$$-3x = 9$$

Step6: Solve for x

Divide by $-3$:
$$x = \frac{9}{-3} = -3$$

Step7: Check validity

The solution $x=-3$ is the restricted value (makes denominator zero), so it is not a valid solution.

Answer:

a. $x=-3$
b. C. The solution set is $\emptyset$